To find the [tex]\( y \)[/tex]-coordinate of the point [tex]\((-7, y)\)[/tex] such that the slope between the points [tex]\((-21, \frac{1}{2})\)[/tex] and [tex]\((-7, y)\)[/tex] is [tex]\(-\frac{1}{7}\)[/tex], we proceed with the following steps:
1. Recall the slope formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
2. Substitute the known values:
- [tex]\( x_1 = -21 \)[/tex]
- [tex]\( y_1 = \frac{1}{2} \)[/tex]
- [tex]\( x_2 = -7 \)[/tex]
- Given slope ([tex]\( m \)[/tex]) = [tex]\(- \frac{1}{7} \)[/tex]
The slope equation becomes:
[tex]\[
-\frac{1}{7} = \frac{y - \frac{1}{2}}{-7 - (-21)}
\][/tex]
3. Simplify the denominator:
[tex]\[
-7 - (-21) = -7 + 21 = 14
\][/tex]
So the slope equation now is:
[tex]\[
-\frac{1}{7} = \frac{y - \frac{1}{2}}{14}
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
- Multiply both sides by 14 to clear the fraction:
[tex]\[
14 \left(-\frac{1}{7}\right) = y - \frac{1}{2}
\][/tex]
- Simplify the left side:
[tex]\[
-2 = y - \frac{1}{2}
\][/tex]
- Add [tex]\(\frac{1}{2}\)[/tex] to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
-2 + \frac{1}{2} = y
\][/tex]
- Convert [tex]\(-2\)[/tex] to a fraction with a common denominator:
[tex]\[
-2 = -\frac{4}{2}
\][/tex]
- Perform the addition:
[tex]\[
-\frac{4}{2} + \frac{1}{2} = -\frac{3}{2}
\][/tex]
Thus, the [tex]\( y \)[/tex]-coordinate of the point [tex]\((-7, y)\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
Therefore, the answer is [tex]\(\boxed{-\frac{3}{2}}\)[/tex].
